\contentsline {chapter}{\numberline {0}基础知识}{1}{chapter.0}%
\contentsline {section}{\numberline {0.1}欧式空间}{1}{section.0.1}%
\contentsline {section}{\numberline {0.2}集合与函数}{1}{section.0.2}%
\contentsline {section}{\numberline {0.3}最优化问题}{2}{section.0.3}%
\contentsline {section}{\numberline {0.4}可微性}{3}{section.0.4}%
\contentsline {section}{\numberline {0.5}线性代数}{4}{section.0.5}%
\contentsline {section}{第 0{} 章\nobreakspace {}练习}{5}{section*.2}%
\contentsline {chapter}{\numberline {1}一维凸函数}{6}{chapter.1}%
\contentsline {section}{\numberline {1.1}定义与例子}{6}{section.1.1}%
\contentsline {section}{\numberline {1.2}基本性质}{8}{section.1.2}%
\contentsline {section}{第 1{} 章\nobreakspace {}练习}{9}{section*.4}%
\contentsline {chapter}{\numberline {2}凸集}{10}{chapter.2}%
\contentsline {section}{\numberline {2.1}凸集的概念}{10}{section.2.1}%
\contentsline {section}{\numberline {2.2}凸集的(代数)运算(保凸操作)}{11}{section.2.2}%
\contentsline {section}{\numberline {2.3}凸组合与凸包}{13}{section.2.3}%
\contentsline {section}{第 2{} 章\nobreakspace {}练习}{15}{section*.5}%
\contentsline {chapter}{\numberline {3}投影}{16}{chapter.3}%
\contentsline {section}{\numberline {3.1}到闭凸集的投影}{16}{section.3.1}%
\contentsline {section}{\numberline {3.2}投影与分离定理}{18}{section.3.2}%
\contentsline {section}{\numberline {3.3}投影计算}{20}{section.3.3}%
\contentsline {section}{\numberline {3.4}应用:随机Kaczmarz算法}{21}{section.3.4}%
\contentsline {section}{第 3{} 章\nobreakspace {}练习}{23}{section*.11}%
\contentsline {chapter}{\numberline {4}锥}{25}{chapter.4}%
\contentsline {section}{\numberline {4.1}锥的概念与性质}{25}{section.4.1}%
\contentsline {section}{\numberline {4.2}Farkas引理与备择定理}{27}{section.4.2}%
\contentsline {section}{\numberline {4.3}线性规划的对偶定理}{28}{section.4.3}%
\contentsline {section}{第 4{} 章\nobreakspace {}练习}{29}{section*.13}%
\contentsline {chapter}{\numberline {5}凸函数}{30}{chapter.5}%
\contentsline {section}{\numberline {5.1}定义与例子}{30}{section.5.1}%
\contentsline {section}{\numberline {5.2}保凸运算}{33}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}保凸的代数运算}{33}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.2.2}保凸的几何运算}{35}{subsection.5.2.2}%
\contentsline {section}{\numberline {5.3}基本性质}{36}{section.5.3}%
\contentsline {section}{第 5{} 章\nobreakspace {}练习}{37}{section*.14}%
\contentsline {chapter}{\numberline {6}次微分}{39}{chapter.6}%
\contentsline {section}{\numberline {6.1}方向导数的性质}{39}{section.6.1}%
\contentsline {section}{\numberline {6.2}次梯度}{40}{section.6.2}%
\contentsline {section}{\numberline {6.3}最速下降方向}{43}{section.6.3}%
\contentsline {section}{\numberline {6.4}次微分法则}{46}{section.6.4}%
\contentsline {section}{第 6{} 章\nobreakspace {}练习}{49}{section*.18}%
\contentsline {chapter}{\numberline {7}共轭函数}{51}{chapter.7}%
\contentsline {section}{\numberline {7.1}定义与基本性质}{51}{section.7.1}%
\contentsline {section}{\numberline {7.2}卷积与共轭}{54}{section.7.2}%
\contentsline {section}{第 7{} 章\nobreakspace {}练习}{57}{section*.20}%
\contentsline {chapter}{\numberline {8}临近算子与Moreau函数}{59}{chapter.8}%
\contentsline {section}{\numberline {8.1}临近点算子}{59}{section.8.1}%
\contentsline {section}{\numberline {8.2}Moreau函数的微分性质}{62}{section.8.2}%
\contentsline {section}{第 8{} 章\nobreakspace {}练习}{65}{section*.23}%
\contentsline {chapter}{\numberline {9}光滑与强凸}{66}{chapter.9}%
\contentsline {section}{\numberline {9.1}基本定义}{66}{section.9.1}%
\contentsline {section}{\numberline {9.2}光滑性的等价刻画}{67}{section.9.2}%
\contentsline {section}{\numberline {9.3}强凸性的等价刻画}{69}{section.9.3}%
\contentsline {section}{\numberline {9.4}光滑强凸性的等价刻画}{70}{section.9.4}%
\contentsline {section}{\numberline {9.5}光滑与强凸之间的对偶}{71}{section.9.5}%
\contentsline {section}{\numberline {9.6}相对光滑与相对强凸条件}{71}{section.9.6}%
\contentsline {section}{第 9{} 章\nobreakspace {}练习}{72}{section*.24}%
\contentsline {chapter}{\numberline {10}误差界条件}{73}{chapter.10}%
\contentsline {section}{\numberline {10.1}强凸松弛条件}{73}{section.10.1}%
\contentsline {section}{\numberline {10.2}典型例子:强凸线性复合函数}{76}{section.10.2}%
\contentsline {section}{第 10{} 章\nobreakspace {}练习}{78}{section*.25}%
\contentsline {chapter}{\numberline {11}最优化条件与对偶}{79}{chapter.11}%
\contentsline {section}{\numberline {11.1}无约束优化情形}{79}{section.11.1}%
\contentsline {section}{\numberline {11.2}约束优化的情形}{81}{section.11.2}%
\contentsline {section}{\numberline {11.3}KKT条件}{84}{section.11.3}%
\contentsline {section}{第 11{} 章\nobreakspace {}练习}{88}{section*.27}%
\contentsline {chapter}{\numberline {12}凸规划的对偶}{89}{chapter.12}%
\contentsline {section}{\numberline {12.1}凸优化问题与条件}{89}{section.12.1}%
\contentsline {section}{\numberline {12.2}备择定理}{89}{section.12.2}%
\contentsline {section}{\numberline {12.3}Lagrange对偶}{90}{section.12.3}%
\contentsline {section}{第 12{} 章\nobreakspace {}练习}{91}{section*.28}%
\contentsline {chapter}{\numberline {13}梯度下降法}{92}{chapter.13}%
\contentsline {section}{\numberline {13.1}问题与算法}{92}{section.13.1}%
\contentsline {section}{\numberline {13.2}线性收敛理论}{96}{section.13.2}%
\contentsline {section}{\numberline {13.3}线性收敛与误差界}{99}{section.13.3}%
\contentsline {chapter}{\numberline {14}加速梯度法}{101}{chapter.14}%
\contentsline {section}{\numberline {14.1}Polyak重球法}{101}{section.14.1}%
\contentsline {subsection}{\numberline {14.1.1}定义}{101}{subsection.14.1.1}%
\contentsline {subsection}{\numberline {14.1.2}理论结果}{102}{subsection.14.1.2}%
\contentsline {section}{\numberline {14.2}Nesterov加速}{104}{section.14.2}%
\contentsline {section}{\numberline {14.3}加速算法的线性收敛}{110}{section.14.3}%
\contentsline {section}{\numberline {14.4}重启加速方法}{112}{section.14.4}%
\contentsline {chapter}{\numberline {15}临近梯度法}{113}{chapter.15}%
\contentsline {section}{\numberline {15.1}算法格式}{113}{section.15.1}%
\contentsline {section}{\numberline {15.2}梯度映射的基本性质}{114}{section.15.2}%
\contentsline {section}{\numberline {15.3}收敛理论}{118}{section.15.3}%
\contentsline {section}{\numberline {15.4}加速的临近梯度法}{120}{section.15.4}%
\contentsline {section}{\numberline {15.5}基于线搜索的临近梯度法}{123}{section.15.5}%
\contentsline {chapter}{\numberline {16}次梯度与条件梯度方法}{128}{chapter.16}%
\contentsline {section}{\numberline {16.1}收敛性分析}{128}{section.16.1}%
\contentsline {section}{\numberline {16.2}收敛率分析}{133}{section.16.2}%
\contentsline {section}{\numberline {16.3}应用举例}{136}{section.16.3}%
\contentsline {section}{\numberline {16.4}推广的格式}{138}{section.16.4}%
\contentsline {section}{第 16{} 章\nobreakspace {}练习}{141}{section*.32}%
\contentsline {chapter}{\numberline {17}条件梯度法}{142}{chapter.17}%
\contentsline {section}{\numberline {17.1}收敛性分析}{142}{section.17.1}%
\contentsline {section}{\numberline {17.2}改进的收敛率}{144}{section.17.2}%
\contentsline {section}{\numberline {17.3}CCCP的条件梯度法解释}{149}{section.17.3}%
\contentsline {section}{第 17{} 章\nobreakspace {}练习}{150}{section*.33}%
